Irreducible

for any two states A and Z, there is some n for which either the probability of reaching Z within n steps of starting from A is nonzero or the probabiltiy of reaching A within n steps of starting from Z is nonzero.

A reducible Markov chain is easily seen by its disconnectedgraph. For instance, if we have 2 states A and B, and P(A->A)=P(B->B)=1, then our graph is 2 disconnected circles, and obviously processes starting at A never read B. In this (extreme) case, any distribution is stable (it's not unique).

If, however, we chose P(A->B)=P(B->B)=1, our chain becomes irreducible, and it has a unique stable distribution: P(B)=1.

Definition Suppose (ρ,V) is a representation of some group G. If there is no proper non-trivial subspace W of V such that ρ(G)W is contained in W, then the representation is said to be irreducible.

Come to think of it, Representation Theory is repsonsible for a great many overloaded definitions; simple is also an equivalent condition to irreducible, then there's complete, regular and characteristic which spring instantly to mind which also have definitions elsewhere in mathematics.
What is nice about the idea of irreducibility is that for finite and indeed compact groups, every representation of the group can be split up into a direct sum of irreducible ones, where the number of such irreducible representations is equal to the number of conjugacy classes of the group.